In the present paper, we generalize these functions to a larger class of elements of PSL2(Z), and explore some of the properties of these maps

Bull. Korean Math. Soc. 50 (2013), No. 2, pp. 601–610 http://dx.doi.org/10.4134/BKMS.2013.50.2.601

A CLASS OF ARITHMETIC FUNCTIONS ON PSL₂(Z)

Paul Spiegelhalter and Alexandru Zaharescu

Abstract. In [3] and [2], Atanassov introduced the two arithmetic func- tions

I(n) = Y

p^α||n

p^1/α and R(n) = Y

p^α||n

p^α−1

called the irrational factor and the restrictive factor, respectively. Alkan, Ledoan, Panaitopol, and the authors explore properties of these arith- metic functions in [1], [7], [8] and [9]. In the present paper, we generalize these functions to a larger class of elements of PSL2(Z), and explore some of the properties of these maps.

1. Introduction

In [3] and [2], Atanassov introduced the two arithmetic functions I(n) = Y

p^α||n

p^1/α and R(n) = Y

p^α||n

p^α−1

called the irrational factor and the strong restrictive factor, respectively. These functions are multiplicative, and satisfy the inequality

I(n)R(n)²≥ n,

with equality if and only if n is squarefree. In [8], Panaitopol showed that X∞

n=1

1

I(n)R(n)φ(n) < e², and proved that the function

G(n) = Yn ν=1

I(ν)^1/n satisfies the inequalities

e⁻⁷n < G(n) < n.

Received December 4, 2011.

2010 Mathematics Subject Classification. Primary 11N37; Secondary 11B57.

Key words and phrases. PSL2(Z), Farey fractions, Dirichlet series.

The second author is supported in part by NSF grant DMS-0901621.

2013 The Korean Mathematical Societyc 601

In [1], Alkan, Ledoan and one of the authors describe a precise asymptotic for the function G(n), and establish further results showing that the function I(n) is very regular on average.

In [7], asymptotic formulas are established for certain weighted real moments of the restrictive factor R(n). In [9], the authors establish asymptotic formulas for weighted combinations I(n)^αR(n)^β.

In the present paper we consider for a matrix A =

 a b c d



in PSL2(Z) the fractional linear transformation Az given by Az = az + b

cz + d. For each positive integer n, define

fA(n) = Y

p^α||n

p^aα+bcα+d.

As an example, the function I(n) is equal to fA0(n) for A0=

 0 1 1 0

 .

We shall consider weighted averages of the functions fA(n). Let MA(x) = 1

x X

n≤x

1 −n x

fA(n).

Consider the subset A ⊂ PSL2(Z) given by A =

 A =

 a b c d



∈ PSL²(Z) : det A = −1, a, b, d ≥ 0, c ≥ 1

 . Define for each positive rational number r

Er= {A ∈ A : MA(x) ≍ x^r as x → ∞}.

Note that if r1 6= r², then Er1 ∩ E^r2 = Ø. We will prove that each Er with r > 0 consists of exactly one element.

For each matrix A in A we define the associated series (Aⁿ)n∈Nby An = An =an + b

cn + d.

As we shall see, the associated series plays an important role in our computa- tions. Clearly, if A ∈ A, then An is monotone decreasing and has the finite limit A∞:= a/c.

We have the following result.

Theorem 1.1. Given A ∈ A, if A¹ > 0, then there are positive real-valued constants KA andc such that

MA(x) = KAx^A1+ OA

x^A1−1/2exp{−c(log x)^3/5(log log x)^−1/5} . We remark that under the Riemann hypothesis, for a restricted class of matrices one has an asymptotic formula for the error term in Theorem 1.1 of the form

(1) MA(x) − K^Ax^A1∼ eKAx^12(A2−1)

for a real-valued constant eKA. This naturally leads one to consider the maps ψj : A → Q⁺ for j = 1, 2 given by

(2) ψj(A) = Aj.

Since, as mentioned above, each Er consists of exactly one element, it follows that there is a well-defined map s : Q+→ Q⁺given by

(3) s(r) = ψ2◦ ψ⁻¹1 (r).

The map s(r) tells us how accurately the main term KAx^A1 approximates MA(x) in (1), in the sense that it gives the exact order of magnitude of the error MA(x) − K^Ax^A1.

Although it can be shown that this map is nowhere continuous, one can obtain asymptotic formulas for the average value of s(r), with r in various ranges. For example, define the height function for each rational r = p/q with q ≥ 1 and (p, q) = 1 by

h(r) := max{|p|, |q|}.

We have the following result.

Theorem 1.2. For any δ > 0, X

r∈Q+∩[0,1]

h(r)≤X

s(r) = 3

2π²X²+ Oδ(X^11/6+δ).

2. Asymptotics of the average Consider the Dirichlet series

FA(s) = X∞ n=1

fA(n) n^s .

We will take advantage of the meromorphic continuation of FA(s) in the case where det A = −1.

Proof of Theorem 1.1. We prove the result with

KA= 1

(1 + A1)(2 + A1)ζ(2)TA(1 + A1).

If det A = −1, then p^Aα≤ p^A1 for all α ≥ 1, so f^A(n) ≤ n^A1, hence FA(s) converges in the half plane ℜs = σ > 1 + A1. Moreover, FA(s) has an Euler product in that region. Write

FA(s) = ζ(s − A¹) ζ(2s − 2A1)

Y

p

(1 + gp(s)) ,

where

gp(s) =

 1 + p^A1

p^s

−1 ∞X

k=2

p^Ak p^ks.

Note that if det(A) = −1, then A¹− A² = (c+d)(2c+d)¹ ≤ ¹2. Take ǫ > 0. For σ ≥ A1+ ǫ we have

 1 + p^A1

p^s

⁻¹

≪ǫ1.

Also, for σ ≥ ¹₂(1 + A2+ ǫ) we have X∞ k=2

p^Ak p^ks ≪ p^A2

p^2s ≪^ǫ 1 p^1+ǫ. Thus for σ ≥ max{A1+ ǫ,¹₂(1 + A2+ ǫ)} the sumP

p|gp(s)| converges, hence TA(s) =Y

p

(1 + gp(s))

is analytic for σ > σ0= max

A1,¹₂(1 + A2)

, so FA(s) is meromorphic there, with a poles at s = 1 + A1.

To continue, we utilize a variant of Perron’s formula and write X

n≤x

1 − n x

fA(n) = 1 2πi

Z c+i∞

c−i∞

ζ(s − A1)

ζ(2s − 2A1)TA(s) x^s s(s + 1)ds, where 1 + A1< c ≤ 5/4 + A1.

We apply the zero-free region for ζ(s) due to Korobov [6] and Vinogradov [12] (see Chapters 2 and 5 of the reference by Walfisz [13] for an alternative treatment)

σ ≥ 1 − c⁰(log t)^−2/3(log log t)^−1/3 for t ≥ t0, in which

1

|ζ(s)| ≪ (log t)^2/3(log log t)^1/3. Fix 0 < U < T ≤ x, let ν = 1/2 + A1 and

η = ν − c0(log U )^−2/3(log log U )^−1/3.

Deform the path of integration into the union of the line segments

















γ1, γ9: s = c + it if |t| ≥ T γ2, γ8: s = σ ± iT if ν ≤ σ ≤ c γ3, γ7: s = ν + it if U ≤ |t| ≤ T γ4, γ6: s = σ ± iU if η ≤ σ ≤ ν γ5: s = η + it if |t| ≤ U.

The integrand is analytic on and within this modified contour, hence by Cauchy’s theorem

xMA(x) = 1

(1 + A1)(2 + A1)ζ(2)TA(1 + A1)x^1+A1+ X9 k=1

Jk,

with the main terms coming from the residue at the simple pole at s = 1 + A1. In order to estimate the integral along our modified contour we will make use of the bounds

|ζ(σ + it)| =







O(t^(1−σ)/2, if 0 ≤ σ ≤ 1 and |t| ≥ 1 O(log t), if 1 ≤ σ ≤ 2

O(1), if σ ≥ 2 (see [11], §3.11 and §5.1).

On the line segments on which s = c + it, |t| ≥ T , we have that ζ(s − A¹) ≪ log t and 1/ζ(2s − 2A¹) ≪ log t, so

|J¹|, |J⁹| ≪ Z ∞

T

(log t)² x^c

|(c + it)(c + 1 + it)|dt

≪ x^c(log T )²

T .

On the line segments on which s = σ + iT , ν ≤ σ ≤ c, we have that 1/ζ(2s − 2A1) ≪ log T , ζ(s − A1) ≪ T^{(1−σ+A1)/2} for ν ≤ σ ≤ 1 + A1, and ζ(s − A1) ≪ log T for 1 + A1≤ σ ≤ c. So

|J²|, |J⁸| ≪

Z 1+A1

ν

T^{12(1−σ+A1)}log Tx^σ T²dσ +

Z c 1+A1

(log T )²x^σ T²dσ

≪ T^12(1+A1)log T maxn  x

√T

ν

,

 x

√T

1+A1o

+(log T )² T² x^c. On the line segments on which s = ν + it, U ≤ |t| ≤ T , we have that ζ(s − A¹) ≪ t^{(1−ν+A1)/2} and 1/ζ(2s − 2A¹) ≪ log t, so

|J3|, |J7| ≪ Z T

U

(log t)t^{12(1−ν+A1)} x^ν

|(ν + it)(ν + 1 + it)|dt

≪ log T U^3/4x^ν.

On the line segments on which s = σ +iU , η ≤ σ ≤ ν, we have that ζ(s−A¹) ≪ U^{(1−σ+A1)/2} and 1/ζ(2s − 2A¹) ≪ log U, so

|J⁴|, |J⁶| ≪ Z ν

η

(log U )U^{12(1−σ+A1)}x^σ U² dσ

≪ U^12(1+A1)−2log U maxn  x

√U

ν

,

 x

√U

ηo .

On the line segment on which s = η + it, |t| ≤ U, we have that ζ(s − A1) ≪ (|t| + 1)^{(1−η+A1)/2}and 1/ζ(2s − 2A¹) ≪ log(|t| + 1), so

|J5| ≪ Z U

−U(|t| + 1)^1−η+A1log(|t| + 1) x^η

|η + it||η + 1 + it| dt

≪ x^η Z U

−U(|t| + 1)^{12(1−η+A1)−2}log(|t| + 1) dt.

Since ¹₂(1 − η + A1) − 2 ≤ −³₂ for U sufficiently large, the above integral converges, hence |J5| ≪ x^η.

We collect all estimates, and take T = x² and

U = exp{c2(log x)^3/5(log log x)^−1/5}

to obtain the desired result. 

One could instead factor

 1 + p^A1

p^s +p^A2 p^2s +p^A3

p^3s + · · ·



=

 1 +p^A1

p^s

  1 +p^A2

p^2s



(1 + gp(s)) with

gp(s) =

 1 + p^A1

p^s

⁻¹ 1 + p^A2

p^2s

⁻¹

−p^A1+A2 p^3s +

X∞ k=3

p^Ak p^ks

!

so that

FA(s) = ζ(s − A1) ζ(2s − 2A¹)

ζ(2s − A2) ζ(4s − 2A²)

Y

p

(1 + gp(s)) .

Under the Riemann hypothesis, we get a second order term of the form eKAx^A2 in the asymptotic formula for FA(s) provided that ¹₄+ A1< ¹₂(1 + A2). That is, provided that

a + b < c + d

2 − 1

2c + d.

This occurs for matrices A in A with restrictions on c and d. One can see that A1will lie in the interval (0, 1/2).

3. Mapping through PSL2(Z) We now return to the two maps ψ1 and ψ2defined in (2).

Lemma 3.1. The map ψ1 is bijective.

Proof. For ^p_q ∈ Q+, consider the set of matrixes A =

 a b c d



in A such that ψ1(A) = ^p_q. We note that any such quadruple (a, b, c, d) is constrained by c ≥ 0, d ≥ 0,

(4) ad − bc = −1

and

(5) c + d = q

p(a + b)

(Note that (4) implies that p cannot be zero). By (5) we have c = q

p(a + b) − d.

Inserting this into (4) gives us

ad − b(a + b)q

p+ bd = −1 so

(a + b)(pd − qb) = −p.

Write a + b = ±n for some positive integer n | p. By (5) we have c + d =

q

p(±n) ∈ Z so p | n, hence p = n.

There are two cases: If a + b = −p, then c + d = −q. This contradicts the assumptions that q ≥ 1 and c and d are non-negative. On the other hand, if a + b = p, then c + d = q, so (4) gives us

a(q − c) − bc = −1 so

(6) pc = 1 + aq.

So c is uniquely determined by cp ≡ 1 (mod q) and 1 ≤ c < q. Then d is uniquely determined by d = q − c, and a and b by a = ^1−pc_q and b = p − a.  In the case where p/q ∈ (0, 1], we identify p/q as an element of FQ, the Farey fractions of order Q, with Q ≥ q. If we consider the “minimal” set of Farey fractions Fq containing p/q, then elementary properties of Farey fractions (see for example Chapter 3 of [5]) give that the adjacent Farey fractions p^′/q^′ <

p/q < p^′′/q^′′ satisfy q^′ = ¯p, p^′ = ¯q, p^′′= p − ¯q and q^′′ = q − ¯p. Here ¯p is the

unique integer 1 ≤ ¯p < q satisfying p¯p ≡ 1 (mod q) and ¯q is the unique integer 1 ≤ ¯q < p satisfying q¯q ≡ 1 (mod p). We can write

ψ1(p/q) =

 q¯ p − ¯q

¯ p q − ¯p

 .

That is, the matrix ψ1(p/q) is comprised of the “parent” Farey fractions in Fq−1.

Additionally, we can write the function s(p/q) from (3) uniquely as

(7) s(p/q) = pp − 1 + pq¯

q(¯p + q) .

To prove Theorem 1.2, we will use the following result (see Lemma 2.3 of [4]).

Lemma 3.2. Assume that q ≥ 1 and h are two given integers, I and J are intervals of length less than q, and f : I × J → R is a C¹ function. Then for any integer T ≥ 1 and any δ > 0

X

a∈I,b∈J ab≡h (mod q)

gcd(a,b)=1

f (a, b) = φ(q) q²

ZZ

I×Jf (x, y)dxdy + E,

with

E ≪δ T²kfk∞q^1/2+δgcd(h, q)^1/2

+ T k∇fk^∞q^3/2+δgcd(h, q)^1/2+k∇fk^∞|I||J |

T ,

wherekfk^∞ andk∇fk^∞denote the sup-norm off and respectively |^∂f∂x| + |^∂f∂y| on I × J .

Proof of Theorem 1.2. Let Q = ⌊X⌋. Since r ∈ FQ we have X

r∈Q+∩[0,1]

h(r)≤X

s(r) = X

1≤q≤Q 1≤p<q (p,q)=1

s(p/q).

We use (7) and Lemma 3.2 with T = q^16−δ3 to get that the right-hand sum is equal to

X

1≤q≤Q

X

1≤p<q 1≤ ¯p<q p ¯p≡1 (mod q)

(p,q)=1

pp − 1 + pq¯

q(¯p + q) = X

1≤q≤Q

φ(q) q²

Z Z

[1,q)²

vu − 1 + uq

q(v + q) dudv + E

= X

1≤q≤Q

φ(q) Z Z

[1/q,1]²

xy − q¹² + x

y + 1 dxdy + E.

where E ≪^δ q^5/6+δ. The integral is equal to 1

2

 1 − 1

q²

  1 −1

q



−q − 1 q³



log 2 − log

 1 + 1

q



= 1 2+ O

1 q



so X

r∈Q+∩[0,1]

h(r)≤X

s(r) = 1 2

X

1≤q≤Q

φ(q) + O



 X

1≤q≤Q

φ(q) q



 + O



 X

1≤q≤Q

q^5/6+δ



 .

One can use the methods of Section 2 to estimate the sums over φ(q), or use partial summation along with standard estimates (see for example [13] or Chapter 18 of [5]). This gives the main term of our theorem; the first error term above is O(X), and the second is Oδ(X^11/6+δ). 

References

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(Kyungshang) 8 (2004), no. 1, 55–58.

[9] P. Spiegelhalter and A. Zaharescu, Strong and weak Atanassov pairs, Proc. Jangjeon Math. Soc. 14 (2011), no. 3, 355–361.

[10] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1939.

[11] , The theory of the Riemann zeta-function, Edited and with a preface by D. R.

Heath-Brown, The Clarendon Press Oxford University Press, New York, 1986.

[12] I. M. Vinogradov, A new estimate of the function ζ(1 + it), Izv. Akad. Nauk SSSR. Ser.

Mat. 22 (1958), 161–164.

[13] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutsche Verlag der Wissenschaften, Berlin, 1963.

Paul Spiegelhalter

Department of Mathematics University of Illinois

1409 West Green Street, Urbana, IL 61801, USA E-mail address: [email protected]

Alexandru Zaharescu Department of Mathematics University of Illinois

1409 West Green Street, Urbana, IL 61801, USA E-mail address: [email protected]